- Research
- Open Access

# ℝ^{3}

- Hu Yan
^{1}Email author

**2012**:3

https://doi.org/10.1186/1029-242X-2012-3

© Yan; licensee Springer. 2012

**Received:**8 September 2011**Accepted:**9 January 2012**Published:**9 January 2012

## Abstract

A special case of Mahler volume for the class of symmetric convex bodies in ℝ^{3} is treated here. It is shown that a cube has the minimal Mahler volume and a cylinder has the maximal Mahler volume for all generalized cylinders. Further, the Mahler volume of bodies of revolution obtained by rotating the unit disk in ℝ^{2} is presented.

**2000 Mathematics Subject Classification:** 52A20; 52A40.

## Keywords

- Mahler volume
- convex body
- polar body
- body of revolution

## 1 Introduction

*K*in Euclidean

*n*-space ℝ

^{ n }is a compact convex set that contains the origin in its interior. Its polar body

*K** is defined by

where *x*·*y* denotes the standard inner product of *x* and *y* in ℝ^{
n
}.

*K*is an origin symmetric convex body, then the product

is called the volume product of *K*, where *V* (*K*) denotes *n*-dimensional volume of *K*, which is known as the *Mahler volume* of *K*, and it is invariant under linear transformation.

One of the main questions still open in convex geometric analysis is the problem of finding a sharp lower estimate for the Mahler volume of a convex body *K* (see the survey article [1]).

*K*in ℝ

^{ n }

with equality if and only if *K* is an ellipsoid centered at the origin, where *ω*_{
n
} is the volume of the unit ball in ℝ^{
n
} (see, e.g., [2–5]).

with equality holding for parallelepipeds and their polars. For *n* = 2, the inequality is proved by Mahler himself [6], and in 1986, Reisner [7] showed that parallelograms are the only minimizers. Reisner [8] established inequality (1.1) for a class of bodies that have a high degree of symmetry, known as zonoids, which are limits of finite Minkowski sums of line segments. Lopez and Reisner [9] proved the inequality (1.1) for *n* ≤ 8 and the minimal bodies are characterized. Recently, Nazaeov et al. [10] proved that the cube is a strict local minimizer for the Mahler volume in the class of origin-symmetric convex bodies endowed with the Banach-Mazur distance.

*c >*0 independent of the dimension

*n*, such that for all origin-symmetric bodies

*K*,

which is now known as the reverse Santaló inequality. Recently, Kuperberg [12] found a beautiful new approach to the reverse Santaló inequality. What's especially remarkable about Kuperberg's inequality is that it provides an explicit value for *c*. However, the Mahler conjecture is still open even in the three-dimensional case, Tao [13] made an excellent remark about the open question.

^{3}. We now introduce some notations: A real-valued function

*f*(

*x*) is called

*concave*, if for any

*x*,

*y*∈ [

*a*,

*b*] and any

*λ*∈ [0, 1], they have

**Definition 1**

*In three-dimensional Cartesian coordinate system OXYZ, if C*′

*is an origin-symmetric convex body in coordinate plane YOZ, then the set:*

*is defined as a generalized cylinder in* ℝ^{3}.

**Definition 2**

*In the coordinate plane XOY, let*

*where f*(

*x*)

*(*[

*-a*,

*a*]

*, a >*0

*), is a nonnegative concave and even function. Rotating D about the X-axis in*ℝ

^{3}

*, we can get a geometric object*

*We define the geometric object R as a body of revolution generated by the function f*(*x*) *(or by the domain D), and call the function f*(*x*) *as the generated function of R and D as the generated domain of R*.

If the generated domain of *R* is a rectangle and a diamond, *R* is called a *cylinder* and a *bicone*, respectively.

Let *C* denotes the set of all generalized cylinders. In this article, we proved that among the generalized cylinders, a cube has the minimal Mahler volume and a cylinder has the maximal Mahler volume, theorem as following:

**Theorem 1**

*For*$C\in \mathcal{C}$,

*we have*

*where C*_{0} = [-1, 1] × [-1, 1] × [-1, 1] *is a cube and C*_{1} = [-1, 1] × *B*^{2} *is cylinder*.

Further, we get the following theorem:

**Theorem 2**

*For a class of bodies of revolution obtained by rotating the "unit disk" in planar XOY, where the "unit disk" is the following set:*

*the Mahler volume is increasing for* 1 ≤ *p* ≤ 2 *and decreasing for* 2 ≤ *p* ≤ +∞.

More interrelated notations, definitions, and their background materials are exhibited in the following section.

## 2 Definition and notation

The setting for this article is *n*-dimensional Euclidean space ℝ^{
n
}. Let ${\mathcal{K}}^{n}$ denotes the set of convex bodies (compact, convex subsets with non-empty interiors), ${\mathcal{K}}_{o}^{n}$ denotes the subset of ${\mathcal{K}}^{n}$ that contains the origin in their interiors. As usual, *B*^{
n
} denotes the unit ball centered at the origin, *S*^{n-1}the unit sphere, *o* the origin, and ||·|| the norm in ℝ^{
n
}.

If *u* ∈ *S*^{n-1}is a direction, *u*^{⊥} is the (*n -* 1)-dimensional subspace orthogonal to *u*. For *x*, *y* ∈ ℝ^{
n
}, *x*·*y* is the inner product of *x* and *y*, and [*x*, *y*] denotes the line segment with endpoints *x* and *y*.

If *K* is a set, ∂*K* is its boundary, *int K* is its interior, and *conv K* denotes its convex hull. *V* (*K*) denotes *n*-dimensional volume of *K*. Let *K*|*S* be the orthogonal projection of *K* into a subspace *S*.

Let $K\in {\mathcal{K}}^{n}$ and *H* = {*x* ∈ ℝ^{
n
}|*x·v* = *d*} denotes a hyperplane, *H*^{+} and *H*^{-} denote the two closed halfspaces bounded by *H*.

*K*in ℝ

^{ n }, its

*support function h*

_{ K }: ℝ

^{ n }

*-*[0, ∞), is defined for

*x*∈ ℝ

^{ n }, by

*radial function ρ*

_{ K }: ℝ

^{ n }

*\*{0} → (0, ∞), is defined for

*x*≠ 0, by

*P*is a polytope, i.e.,

*P*= conv{

*p*

_{1}, ...,

*p*

_{ m }}, where

*p*

_{ i }(

*i*= 1, ...,

*m*) are vertices of polytope

*P*. By the definition of polar body, we have

which implies that *P** is the intersection of *m* closed halfspace with exterior normal vector *p*_{
i
} and the distance of hyperplane {*x* ∈ ℝ^{
n
} : *x*·*p*_{
i
} = 1} from the origin is 1*/*||*p*_{
i
}||.

For $K\in {\mathcal{K}}_{o}^{n}$, if *x* = (*x*_{1}, *x*_{2}, ..., *x*_{
n
}) ∈ *K*, *x*′ = (*ε*_{1}*x*_{1}, ..., *ε*_{
n
}*x*_{
n
}) ∈ *K* for any signs *ε*_{
i
} = ± 1 (*i* = 1, ..., *n*), then *K* is a *1-unconditional convex body*. In fact, *K* is symmetric around all coordinate hyperplanes.

To proof the inequality, we give the following definitions.

**Definition 3**

*In Definition 2, if the function*

*where k and b are real constants, and f*(

*-a*) = 0

*, then the body of revolution is defined as a bicone. In three-dimensional Cartesian coordinate system OXYZ, if C*′

*is an origin-symmetric convex body in coordinate plane YOZ and points A*= (

*-a*, 0, 0)

*and A*′ = (

*a*, 0, 0)

*, then the set*.

*is defined as a generalized bicone in* ℝ^{3}.

## 3 Proof of the main results

In this section, we only consider convex bodies in three-dimensional Cartesian coordinate system with origin *O*, and its three coordinate axes are denoted by *X*-, *Y* -, and *Z*-axis.

*C*be a generalized cylinder as following:

where *C*′ is an origin-symmetric convex body in coordinate plane *Y OZ*.

We require the following lemmas to prove our main result.

**Lemma 1**

*If*$K\in {\mathcal{K}}_{o}^{3}$

*, for any u*∈

*S*

^{2}

*, then*

*On the other hand, if*${K}^{\prime}\in {\mathcal{K}}_{o}^{3}$

*satisfies*

*for any*$u\in {S}^{2}\cap {v}_{0}^{\perp}$ (

*v*

_{0}

*is a fixed vector*)

*, then*

**Proof** First, we prove (3.1).

*x*∈

*u*

^{⊥},

*y*∈

*K*and

*y*′ =

*y*|

*u*

^{⊥}, since the hyperplane

*u*

^{⊥}is orthogonal to the vector

*y*-

*y*′, then

*x*∈

*K** |

*u*

^{⊥}, for any

*y*′ ∈

*K*|

*u*

^{⊥}, there exists

*y*∈

*K*such that

*y*′ =

*y*|

*u*

^{⊥}, then

*x·y*′ =

*x*·

*y*≤ 1, and

*x*∈ (

*K*|

*u*

^{⊥})*. Hence,

*x*∈ (

*K*|

*u*

^{⊥})*, then for any

*y*∈

*K*and

*y*′ =

*y*|

*u*

^{⊥},

*x*·

*y*=

*x*·

*y*′

*≤*1, thus

*x*∈

*K**, and since

*x*∈

*u*

^{⊥}, thus

*x*∈

*K** |

*u*

^{⊥}. Then,

Next, we prove (3.2).

*v*∈

*S*

^{2}, there always exists a

*u*∈

*S*

^{1}satisfying

*v*∈

*u*

^{⊥}. Since

By the arbitrary of direction *v*, we obtain the desired result. ■

*u*∈

*B*

^{2}|

*v*

^{⊥}(

*v*= (1, 0, 0)),

*C*|

*u*

^{⊥}is a rectangle by the above definition. We study the polar body of a rectangle in the planar. From Figure 1, if

*C*|

*u*

^{⊥}= [-1, 1]

*×*[

*-a*,

*a*], its polar body in planar

*XOY*is a diamond (vertices are (-1, 0), (1, 0), (0, -1

*/a*), (0, 1

*/a*)), thus we can get the following Lemma 2.

**Lemma 2**

*For any*$C\in \mathcal{C}$

*, if*

*where C*′ *is an origin-symmetric convex body in coordinate plane YOZ, then C** *is a generalized bicone with vertices* (-1, 0, 0) *and* (1, 0, 0) *and the base* (*C*′)*.

**Proof**Let

*v*

_{0}= (1, 0, 0), ${S}^{1}={S}^{2}\cap {v}_{0}^{\perp}$. By Lemma 1, we have

for any *u* ∈ *S*^{1}. Because that (*C*|*u*^{⊥})* is a diamond with vertices (-1, 0, 0) and (1, 0, 0), *C** | *u*^{⊥} is a diamond with vertices (-1, 0, 0) and (1, 0, 0) for any *u* ∈ *S*^{1}, which implies that *C** is a bicone with vertices (-1, 0, 0) and (1, 0, 0).

and $C|{v}_{0}^{\perp}={C}^{\prime}$, then, the base of *C** is (*C*′)*. ■

In the following, we will restate and prove Theorem 1.

**Theorem 1**

*For*$C\in \mathcal{C}$

*, we have*

*where C*_{0} = [-1, 1] *×* [-1, 1] *×* [-1, 1] *is a cube and C*_{1} = [-1, 1] *× B*^{2} *is cylinder*.

**Proof** Let *v* = (1, 0, 0), and *V* (*C*) = *V* (*C*_{0}) by linear transformation, thus *V* (*C* ∩ *v*⊥) = *V* (*C*_{0} ∩ *v*^{⊥}).

*v*

^{⊥}, since the square has the minimal Mahler volume in ℝ

^{2}, thus

*C*∩

*v*

^{⊥}is a square. Hence,

Similarly, let *V* (*C*) = *V* (*C*_{1}) for any $C\in \mathcal{C}$ by linear transformation, then *V* (*C* ∩ *v*^{⊥}) = *V* (*C*_{1} ∩ *v*^{⊥}).

*C*

_{1}∩

*v*

^{⊥}is a disk, which has the maximal Mahler volume in ℝ

^{2}, thus

■

Theorem 1 implies that among the generalized cylinders, a cube has the minimal Mahler volume and a cylinder has the maximal Mahler volume.

## 4 Mahler volume of a special class of bodies of revolution

*XOY*, and define the "unit disk" in planar

*XOY*as following set:

We need the following lemmas to prove our result.

**Lemma 3** *Let P is a 1-unconditional convex body and P** *is its polar body in the coordinate plane XOY. Let R and R*′ *are two bodies of revolution obtained by rotating P and P***, respectively. Then R*′ = *R**.

**Proof**Let

*v*

_{0}= {1, 0, 0} and ${S}^{1}={S}^{2}\cap {v}_{0}^{\perp}$, for any

*u*∈

*S*

^{1}, we have

*R*′ ∩

*u*⊥ = (

*R*∩

*u*

^{⊥})* for any

*u*∈

*S*

^{1}, we get

for any *u* ∈ *S*^{1}. By Lemma 1, we have *R*′ = *R**. ■

**Lemma 4**

*If*

*then the polar body of*

*is the following set:*

**Proof**For any (

*x*,

*y*) ∈

*U*and (

*x*′,

*y*′) ∈

*U*′, we have

which implies *U*′ ⊂ *U**.

*A*′ = (

*x*′,

*y*′) ∉

*U*′, then

*r >*1 and a point

*A*

^{0}∈ ∂

*U*′ satisfying ${A}_{0}^{\prime}=r{A}^{0}$. If

*A*

^{0}= (

*x*

_{0},

*y*

_{0}), then

*x*

_{0}

*>*0 and

*y*

_{0}

*>*0. Let $x={x}_{0}^{\frac{q}{p}}$ and $y={y}_{0}^{\frac{q}{p}}$, then

which implies (*x*, *y*) ∈ *U* and 〈(*x*, *y*), (|*x*′|, |*y*′|)〉 = *r >* 1, thus ${A}_{0}^{\prime}\notin {U}^{*}$. Because that *U** is a 1-unconditional convex body, we have *A*′ ∉ *U**. Then, *U** ⊂ *U*′. ■

Rotating *U* and *U*′, we can get two bodies of revolution *R* and *R*′. By Lemma 3, we have *R*′ = *R**. Let *F*(*p*) = *V* (*R*)*V* (*R**).

In the following, we restate and prove Theorem 2.

**Theorem 2**

*For a class of bodies of revolution obtained by rotating the "unit disk" in planar XOY, where the "unit disk" is the following set:*

*the Mahler volume is increasing for* 1 ≤ *p* ≤ 2 *and decreasing for* 2 ≤ *p* ≤ +∞.

**Proof**By integration, we get

*V*

_{ R }(

*p*) and

*V*

_{R*}(

*q*), which are volume functions of

*R*and

*R** about

*p*and

*q*as following:

*V*(

*R*)

*V*(

*R**), which is a function about

*p*as following:

where *p* ≥ 1, and $\frac{1}{p}+\frac{1}{q}=1$.

*x*

^{ p }=

*y*, we have

*B*(·, ·) is Beta function. Thus we have

where *p* ≥ 1, and $\frac{1}{p}+\frac{1}{q}=1$.

*R*and

*R** are bicone and cylinder, or cylinder and bicone, and

then *R* and *R** are the same unit ball, which have the maximal Mahler volume.

In fact, *F*(*p*) = *F*(*q*) holds when $\frac{1}{p}+\frac{1}{q}=1$, so we just need to prove *F* (*p*) is increasing when 1 ≤ *p* ≤ 2, which can be easily proved by *F*′(*p*) ≥ 0 when 1 ≤ *p* ≤ 2. Based on the above conclusions, we have that a cylinder has the minimal Mahler volume and a ball has the maximal Mahler volume in this special class of bodies of revolution. ■

*F*(

*p*) by using MATLAB (see Figure 2). From the figure, we see that function

*F*(

*p*) is increasing when 1 ≤

*p*≤ 2 and decreasing when 2 ≤

*p*≤ +∞, so

*F*(2) is a maximum and

*F*(1) =

*F*(+∞) is a minimum.

## Declarations

### Acknowledgements

The author express her deep gratitude to the referees for their many very valuable suggestions and comments. The research of Hu-Yan was supported by National Natural Science Foundation of China (10971128), Shanghai Leading Academic Discipline Project (S30104), and the research of Hu-Yan was partially supported by Innovation Program of Shanghai Municipal Education Commission (10yz160).

## Authors’ Affiliations

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